Select the arrow sign (which means "next") at the bottom of the screen after taking a picture. You can check the picture/video you have taken along with the background picture and status message of your profile.

Click "Apply" and the picture will be saved as your profile picture. If the new photo is not shown, please check again if your KakaoTalk is the latest version.

There are two options.
1. Go to KakaoTalk > Edit Profile. Select "profilecons" from the Profile menu, and you will be directed to the relevant page where you can choose new profilecons.
2. Select "Gallery" on the top left of the camera screen, and you will see a button right next to your profile picture which will direct you to the profilecon page.